System modeling Lesson 3 Systems and Control Theory STADIUS - - - PowerPoint PPT Presentation

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

System modeling

Lesson 3

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Introduction

2

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Modeling of dynamical systems We can derive the mathematical model of a system in two ways mainly:

• Physical Modeling

It consists of applying various laws of physics, chemistry, thermodynamics, etc., to derive ODE or PDE models. It is modeling from “First Principles”.

Mass-spring system Inverted pendulum Tubular chemical reactor

r r w

( )

E RT E RT

C C v k Ce t z T T v G Ce H T T t z

 

               

J1

T A B  

J2

T

J3

T

in in

, C T reactant , C T A reactant A product B

Systems and Control Theory

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Introduction

3

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• System identification or Empirical Modeling

It consists of developing models from observed or collected data.

Dynamical system

1( )

u t

2( )

u t ( )

n

u t

1( )

y t

2( )

y t ( )

m

y t

Identification Algorithm

Tuning parameters

s

T

s

T

Mathematical Model  

( ) ( 1), ( 2), , ( ), ( 1), t t t t t     y f y y u u

For example:

Systems and Control Theory

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Physical modeling

Systems and Control Theory

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Example: Mass Spring System

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• Dynamical system

Systems and Control Theory

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Mass Spring Damper System

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Systems and Control Theory

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Mass Spring Damper System

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https://www.youtube.com/watch?v=8DuJEpy-ODo

Systems and Control Theory

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Example: Pendulum

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Inverted pendulum

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Systems and Control Theory

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Flying Inverted Pendulum

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https://www.youtube.com/watch?v=15DIidigArA

Systems and Control Theory

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Example: LC circuit

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Systems and Control Theory

STADIUS - Center for Dynamical Systems,

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Example: RLC circuit

• Equations for each component
• Let V2 and i be the states.

(They are already in the derivative).

• Model output as function of

states and inputs

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Systems and Control Theory

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Example: linear circuit

• Equations from previous slide
• Writing the equations as

matrices results in state space representation

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Systems and Control Theory

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Force-Voltage Analogy

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Systems and Control Theory

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Force-Voltage Analogy

• Force F
• Mass m
• Viscous-friction coefficient b
• Spring constant k
• Displacement
• Velocity

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• Voltage e
• Inductance L
• Resistance R
• Reciprocal of capacitance 1/C
• Charge q
• Current i

Systems and Control Theory

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Example: Hoover dam

• :inflow of water in
• :current volume of water

in

• :outflow to river in
• :the current water level

(height) in

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Systems and Control Theory

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Example: Hoover dam

• What happens when we open the gate?
• Outflow (like a brick of milk)
• We assume (falsely) that

depends linearly on .

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